# addition and subtraction formulas for sine and cosine

The rotation matrix $\displaystyle\left(\begin{array}[]{lr}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)$ will be used to obtain the addition formulas for sine and cosine.

Recall that a vector in $\mathbb{R}^{2}$ can be rotated $\theta$ radians in the counterclockwise direction by multiplying on the left by the rotation matrix $\displaystyle\left(\begin{array}[]{lr}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)$. Because rotating by $\alpha+\beta$ radians is the same as rotating by $\beta$ radians followed by rotating by $\alpha$ radians, we obtain:

$\begin{array}[]{rl}\displaystyle\left(\begin{array}[]{lr}\cos(\alpha+\beta)&-% \sin(\alpha+\beta)\\ \sin(\alpha+\beta)&\cos(\alpha+\beta)\end{array}\right)&=\displaystyle\left(% \begin{array}[]{lr}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{lr}\cos\beta&-% \sin\beta\\ \sin\beta&\cos\beta\end{array}\right)\\ &\\ &=\displaystyle\left(\begin{array}[]{lr}\cos\alpha\cos\beta-\sin\alpha\sin% \beta&-\cos\alpha\sin\beta-\sin\alpha\cos\beta\\ \sin\alpha\cos\beta+\cos\alpha\sin\beta&-\sin\alpha\sin\beta+\cos\alpha\cos% \beta\end{array}\right)\end{array}$

Hence, $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$ and $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$.

Note that sine is an even function and that cosine is an odd function, i.e. (http://planetmath.org/Ie) $\sin(-x)=-\sin x$ and $\cos(-x)=-\cos x$. These facts enable us to obtain the subtraction formulas for sine and cosine.

 $\sin(\alpha-\beta)=\sin(\alpha+(-\beta))=\sin(\alpha)\cos(-\beta)+\cos(\alpha)% \sin(-\beta)=\sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)$
 $\cos(\alpha-\beta)=\cos(\alpha+(-\beta))=\cos(\alpha)\cos(-\beta)-\sin(\alpha)% \sin(-\beta)=\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)$
 Title addition and subtraction formulas for sine and cosine Canonical name AdditionAndSubtractionFormulasForSineAndCosine Date of creation 2013-03-22 16:59:01 Last modified on 2013-03-22 16:59:01 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 16 Author Wkbj79 (1863) Entry type Derivation Classification msc 26A09 Classification msc 15-00 Classification msc 33B10 Synonym addition and subtraction formulae for sine and cosine Synonym addition formulas for sine and cosine Synonym addition formulae for sine and cosine Synonym subtraction formulas for sine and cosine Synonym subtraction formulae for sine and cosine Synonym addition formula for sine Synonym subtraction Related topic AdditionFormula Related topic DefinitionsInTrigonometry Related topic DoubleAngleIdentity Related topic MeanCurvatureAtSurfacePoint Related topic DAlembertAndDBernoulliSolutionsOfWaveEquation Related topic AdditionFormulas